11.) [And fourth write up.] Let
M =
n2n ? 1
4n + 1
n 2 N
o
:
a. Determine the limit point of M and prove that it is a limit
point.
b. Determine the least upper bound of M and prove that it is the
least
upper bound.![11.) [And fourth write up.] Let 2n M = An + } 1 1 a. Determine the limit point of M and prove that it is a limit point. b. De](http://img.homeworklib.com/questions/88a0bac0-dff9-11ea-b90c-e13942ea4253.png?x-oss-process=image/resize,w_560)
Homework Answers
4. For the following sets determine the least upper bound (it is not necessary to prove...
4. For the following sets determine the least upper bound (it is
not necessary
to prove that it is the least upper bound):
a.) M = [0; 1] [ (3; 4)
b.) M =
n5n + 1
4n ? 3
n 2 N
o
c.) M =
n n + 1
2n + 13
n 2 N
o
d.) M =
nXn
i=1
9
10i
n 2 N
o
e.) M =
n
xjx > 0 and x2 < 5g:...
Some useful identities Using (2.3), we have n2n-1 n2n-1 + n(n 1)2"-2 non-1 + n(n-1)(n-2)2n-3 +...
Some useful identities Using (2.3), we have n2n-1 n2n-1 + n(n 1)2"-2 non-1 + n(n-1)(n-2)2n-3 + 3n(n-1)2n-2 7n İfp-3 n- 22n(n1) if p 2 2"-3n2(n +3) if p3 Using (2.4) and (2.5) we have 0 ifpe(0, 1, ,n-1} Can you give combinatorial explanations for these identities?
. 1. Prove by induction that for all integers n≥1, 4+8+12+...+4n = 2n^2+2n 2. A number...
. 1. Prove by induction that for all integers n≥1, 4+8+12+...+4n = 2n^2+2n 2. A number a is divisible by b if the remainder of dividing a by b is zero. For example 10 is divisible by 5 but 11 is not divisible by 5. Prove by induction that for all integers n≥1,11^n - 6 is divisible by 5. 3. Prove by induction that for all integers n ≥ 1, 3^n ≥ 2^n+n^2
12. Determine whether the following series converge or diverge. (a) (b) 2-nzn-1 4n n=0 n=1 4n...
12. Determine whether the following series converge or diverge. (a) (b) 2-nzn-1 4n n=0 n=1 4n (-1)n+1 loge n (c) (d) 7n + 1 n n=1 n=3 iM: M: Mį M8 sinn (e) ✓n n2 + 2 (f) n2 n=1 n=1 2n en (g) (h) Vn! n=1 n=1
use a bijective argument 1 k/n) m-1 Prove that n2n-l-Li
use a bijective argument
1 k/n) m-1 Prove that n2n-l-Li
2. (a) Let 11 = 0 and Zn+1=2r" +1 for all n E N. In +2...
2. (a) Let 11 = 0 and Zn+1=2r" +1 for all n E N. In +2 i. Find 2, , and ii. Prove that (r converges and find the value of its limit (b) Let a-V2, and define @n+1 = V2+@n for all n 1. Prove that lim an exists and equals 2 Hint: For both parts try to apply the Monotone Convergence Theorem
7. Let E C R be nonempty, n E N, and K, L E Z such that K/n is an upper bound for E, but L/n is not an upper bound for...
7. Let E C R be nonempty, n E N, and K, L E Z such that K/n is an upper bound for E, but L/n is not an upper bound for E. (a) Show that there exists an for E, but (m - 1)/n is not an upper bound for E. (Hint: Prove by contradiction, and use induction. Drawing a picture might help) m < K such that m/n is an upper bound integer L (b) Show that m...
1. a) Let f(n) = 6n2 - 100n + 44 and g(n) = 0.5n3 . Prove...
1. a) Let f(n) = 6n2 - 100n + 44 and g(n) =
0.5n3 . Prove that f(n) = O(g(n)) using the definition
of Big-O notation. (You need to find constants c and n0).
b) Let f(n) = 3n2 + n and g(n) = 2n2 . Use
the definition of big-O notation to prove that
f(n) = O(g(n)) (you need to find constants c and n0) and
g(n) = O(f(n)) (you need to find constants c and n0).
Conclude that...
PLEASE SHOW WORK Question 3 Use mathematical induction to prove 3+7+11+ ... +(4n – 1) =...
PLEASE SHOW WORK
Question 3 Use mathematical induction to prove 3+7+11+ ... +(4n – 1) = n (2n + 1). • Show P1 is true. • Assume Pk is true. • Show Pk+1 is true.
Use the definition of 0 to show that 5n^5 +4n^4 + 3n^3 + 2n^2 + n...
Use the definition of 0 to show that 5n^5 +4n^4 + 3n^3 + 2n^2 + n 0(n^5).Use the definition of 0 to show that 2n^2 - n+ 3 0(n^2).Let f,g,h : N 1R*. Use the definition of big-Oh to prove that if/(n) 6 0(g{n)) and g(n) 0(h{n)) then/(n) 0(/i(n)). You should use different letters for the constants (i.e. don't use c to denote the constant for each big-Oh).
4. For the following sets determine the least upper bound (it is
not necessary
to prove that it is the least upper bound):
a.) M = [0; 1] [ (3; 4)
b.) M =
n5n + 1
4n ? 3
n 2 N
o
c.) M =
n n + 1
2n + 13
n 2 N
o
d.) M =
nXn
i=1
9
10i
n 2 N
o
e.) M =
n
xjx > 0 and x2 < 5g:...
Some useful identities Using (2.3), we have n2n-1 n2n-1 + n(n 1)2"-2 non-1 + n(n-1)(n-2)2n-3 + 3n(n-1)2n-2 7n İfp-3 n- 22n(n1) if p 2 2"-3n2(n +3) if p3 Using (2.4) and (2.5) we have 0 ifpe(0, 1, ,n-1} Can you give combinatorial explanations for these identities?
12. Determine whether the following series converge or diverge. (a) (b) 2-nzn-1 4n n=0 n=1 4n (-1)n+1 loge n (c) (d) 7n + 1 n n=1 n=3 iM: M: Mį M8 sinn (e) ✓n n2 + 2 (f) n2 n=1 n=1 2n en (g) (h) Vn! n=1 n=1
use a bijective argument
1 k/n) m-1 Prove that n2n-l-Li
2. (a) Let 11 = 0 and Zn+1=2r" +1 for all n E N. In +2 i. Find 2, , and ii. Prove that (r converges and find the value of its limit (b) Let a-V2, and define @n+1 = V2+@n for all n 1. Prove that lim an exists and equals 2 Hint: For both parts try to apply the Monotone Convergence Theorem
7. Let E C R be nonempty, n E N, and K, L E Z such that K/n is an upper bound for E, but L/n is not an upper bound for E. (a) Show that there exists an for E, but (m - 1)/n is not an upper bound for E. (Hint: Prove by contradiction, and use induction. Drawing a picture might help) m < K such that m/n is an upper bound integer L (b) Show that m...
1. a) Let f(n) = 6n2 - 100n + 44 and g(n) =
0.5n3 . Prove that f(n) = O(g(n)) using the definition
of Big-O notation. (You need to find constants c and n0).
b) Let f(n) = 3n2 + n and g(n) = 2n2 . Use
the definition of big-O notation to prove that
f(n) = O(g(n)) (you need to find constants c and n0) and
g(n) = O(f(n)) (you need to find constants c and n0).
Conclude that...
PLEASE SHOW WORK
Question 3 Use mathematical induction to prove 3+7+11+ ... +(4n – 1) = n (2n + 1). • Show P1 is true. • Assume Pk is true. • Show Pk+1 is true.
Use the definition of 0 to show that 5n^5 +4n^4 + 3n^3 + 2n^2 + n 0(n^5).Use the definition of 0 to show that 2n^2 - n+ 3 0(n^2).Let f,g,h : N 1R*. Use the definition of big-Oh to prove that if/(n) 6 0(g{n)) and g(n) 0(h{n)) then/(n) 0(/i(n)). You should use different letters for the constants (i.e. don't use c to denote the constant for each big-Oh).
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